Towards
a Non-hydrostatic Spectral Model

Takeshi Enomoto

Kyoto University

Abstract:The Earth Simulator ignited high-resolution global
simulations. We
conducted 10-km simulations using AFES (AGCM for the Earth Simulator)
at T1279L96 in 2002 (Ohfuchi et al. 2004). Although we believe it was a
landmark in the history of AGCM, it seemed to me that T1279 hit the
limit of hydrostatic spectral models. In fact other groups started
developing a non-hydrostatic grid-point models such as NICAM (Tomita
and Satoh 2004) to permit explicit cumulus convection and to avoid
computationally demanding Legendre transforms. While I have enjoyed
seeing development of non-hydrostatic models, I have always felt that
spectral models can also be improved. In this seminar I will discuss my
contributions with collaborators to keep spectral models competitive in
the non-hydrostatic regime. They are 1) accurate methods to calculate
the associated Legendre functions (ALF), 2) an accurate and simple
interpolation scheme and 3) simple and stable non-hydrostatic
formulation.

ALF is the prerequisite for the accuracy of the
Legendre
transform thus of the dynamical core. Enomoto et al. 2008) show that
ALF calculated with the common three-point recurrence fails at high
degree and order and recommend the alternative four-point recurrence
(Swarztrauber 1993). Although the latter enables stable calculation at
high order and degree, the accuracy of each value of ALF has not been
confirmed. Recently Fukushima (2011) proposed an efficient and accurate
method to calculate ALF with the conventional three-point recurrence
using the extended floating-point arithmetic. ALF calculated with the
four-point recurrence are verified with those calculated with
Fukushima’s method. ALF calculated with the four-point recurrence is
accurate to about 13 digits below the floating point but are inaccurate
at very small values (say <10!!"). Although these small values
do not affect the accuracy of the Legendre transform, Fukusima’s method
is a good replacement due to its capability to calculate wide range of
values and to parallelize in the zonal wave number m.

Majority of
weather forecast and some climate models uses semi-Lagrangian advection
in favour of a longer time step. In addition it eliminates dispersion,
which is unavoidable with Eulerian advection. The problem of
semi-Lagrangian advection is dissipation inherent in interpolation.
Enomoto (2008) proposes the use of the spectral derivatives in the
bicubic interpolation. He conducted rotation of a Gaussian hill
(Ritchie 1987) with various interpolation methods and demonstrated the
advantages of the new method. Although the advective form does not
formally conserve mass, dissipation is very small with this
interpolation. Recently a standard suite of advection tests has been
proposed (Lauritzen et al. 2012). It includes smooth and non-smooth
tracers under deformational non-divergent and divergent flows. The
proposed interpolation is verified with the standard tests and is found
to be very competitive with other state-of-art advection schemes.

The
rapid increase of computing power is making global non-hydrostatic
simulations affordable. A natural approach is to extend the formulation
to include the non-hydrostatic effect. The advantage of this approach
is that the existing data assimilation and tools require minimal
changes. ECMWF and JMA seem to pursue this approach. ECMWF has achieved
TL7999 (corresponding to approximately 2.5 km) with a fast Lendre
transform using the butterfly algorithm (N. Wedi, pers. comm.).
Hiromasa Yoshimura (MRI/JMA) has built a non-hydrostatic version of JMA
GSM using double Fourier series. Their formulations are based on
Laprise 1992) that proposes the vertical co-ordinates based on
hydrostatic pressure. Juang (1992, 2000) also adopts hydrostatic
℀–co-ordinates in the vertical but there are subtle differences. The
latter introduces the hydrostatic temperature. In a limited-area model,
such as MSM, the hydrostatic temperature may be given xternally. In a
GCM, however, the hydrostatic temperature must be determined internally
if is not time-independent. I investigated the two formations and found
the assumption of the hydrostatic state of Laprise (1992) may be used
to diagnose the hydrostatic temperature within MSM. Similarly the
hydrostatic assumption of Arakawa and Konor (2009) can be used. MSM is
found to run stably with any of these diagnosed hydrostatic states. The
diagnosed hydrostatic temperature would enable the application of the
formulation of MSM to the global domain